Radon Transform Imaging Aswin C. Sankaranarayanan Good afternoon, every one. The paper I am going to talk about is a novel imaging architecture for video compressive sensing Jian Wang Mohit Gupta

light transport/scattering media multi-pixel multiplexing cameras Compressive Imaging single pixel camera (Duarte et al. 2007) CASSI (Gehm et al. 2007, Wagadarikar et al. 2008) High-speed imaging (Reddy et al. 2011, Hitomi et al. 2011, Holloway et al. 2012) light transport/scattering media (Gu et al. 2008, Peers et al. 2008, S. 2009, Sen and Darabi 2009) multi-pixel multiplexing cameras (Mahalanobis et al. 2014, Kerviche et al. 2014, Wang et al. 2015, Chen et al. 2015)

Spatial Multiplexing Optically super-resolve a low-resolution sensor using a spatial light modulator Table courtesy of Gehm and Brady, Applied Optics, 2015

Spatial Multiplexing ~ ~ ~ Optically super-resolve a low-resolution sensor using a spatial light modulator objective lens digital micromirror device (DMD) ~ ~ ~ photo- detector relay lens ADC

Low-resolution sensor Spatial Multiplexing Optically super-resolve a low-resolution sensor using a spatial light modulator objective lens relay lens ~ ~ ~ digital micromirror device (DMD) Low-resolution array ADC Low-resolution sensor DMD

Key Limitations Achievable spatial resolution is limited to that of the DMD 2-4 megapixels today At least, half-the-light is lost due to spatial light modulator Extended optical axis leads to vignetting and/or blur objective lens relay lens ~ ~ ~ digital micromirror device (DMD) Low-resolution array ADC

Key Limitations ~ ~ ~ Benefits of CS are magnified at high-resolutions Achievable spatial resolution is limited to that of the DMD 2-4 megapixels today At least, half-the-light is lost due to spatial light modulator Extended optical axis leads to vignetting and/or blur Benefits of CS are magnified at high-resolutions objective lens relay lens ~ ~ ~ digital micromirror device (DMD) Low-resolution array ADC

Sparsity of natural images 10 megapixel image

Sparsity of natural images keep largest K coefficients, set others to zero wavelet transform reconstruct the image N pixels, N coefficients K - # of non-zero coefficients K/N – non-zero ratio

Sparsity of natural images Reconstruction SNR [dB] Non-zero ratio K/N N [in pixels] 1E4 1E5 1E6 1E7 0.01 0.1 128128 1 MP 20 MP SNR > 25 dB K/N = 0.1 35 30 25 20 K/N = 0.01 1E4 1E5 1E6 1E7 128x128 1MP 10MP N [in pixels] K - # of non-zero coefficients N - # of pixels

We need a light-modulator free design for compressive imaging Key Limitations Achievable spatial resolution is limited to that of the DMD 2-4 megapixels today At least half-the-light is lost Long optical chain leads to vignetting and/or blur Benefits of CS are magnified at high-resolutions We need a light-modulator free design for compressive imaging

Radon Transform Imaging Tomographic measurements Key idea underlying many medical imaging techniques including CT, MRI, etc. Key question: Can we achieve tomographic measurements optically (for imaging in visible/infrared) ?

Optical Setup rotation stage objective lens line-sensor cylindrical axis line-sensor cylindrical lens rotation stage

Optical Setup z z x y Objective lens Cylindrical lens Line Sensor (focal length ) Line Sensor scene plane z x Line Sensor scene plane z y Objective lens Cylindrical lens (focal length )

Optical Setup rotation stage Key properties axis objective lens cylindrical lens rotation stage line-sensor Key properties (i) Focuses light in the direction along the line-sensor (ii)Defocuses light in the direction perpendicular to the line-sensor

Radon Transform Imaging Recovering an image (or a video) is by solving a linear inverse problem Many efficient solvers thanks to advances in CS and tomography/MRI. x – image y – measurements A – measurement operator Ψ – sparsifying prior image x measurement y

Benefits of Radon Transform Imaging Given a line-sensor with N pixels, we can (in theory) achieve a spatial resolution of NxN Hence, capable of achieving extremely high resolutions

Benefits of Radon Transform Imaging Given a line-sensor with N pixels, we can (in theory) achieve a spatial resolution of NxN Hence, capable of achieving extremely high resolutions Fast implementation (via Fourier Slice Theorem)

Benefits of Radon Transform Imaging Given a line-sensor with N pixels, we can (in theory) achieve a spatial resolution of NxN Hence, capable of achieving extremely high resolutions Fast implementation (via Fourier Slice Theorem) No light loss Requires careful design of optical elements No long light paths (no vignetting etc) Limitations: Moving components

256x256 17.7 dB 10x Comp.

512x512 20.4 dB 10x Comp.

1024x1024 22.1 dB 10x Comp.

2048x2048 23.4 dB 10x Comp.

Summary A spatial light modulator free design for imaging in visible/IR Hence, capable of achieving extremely high resolutions Has many desirable properties Co-axial components Little light loss with careful design of optical elements Fast implementation via the Fourier transform